Almost prime

In number theory, a natural number is called almost prime if there exists an absolute constant K such that the number has at most K prime factors.[1][2] An almost prime n is denoted by P_r if and only if the number of prime factors of n, counted according to multiplicity, is at most r.[3] A natural number is called k-almost prime if it has exactly k prime factors, counted with multiplicity. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the prime factorization of n (can be also seen as the sum of all the primes' exponents):

${\displaystyle \Omega (n):=\sum a_{i}\qquad {\mbox{if}}\qquad n=\prod p_{i}^{a_{i}}.}$

Demonstration, with Cuisenaire rods, of the 2-almost prime nature of the number 6

A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of k-almost primes is usually denoted by P_k. The smallest k-almost prime is 2^k. The first few k-almost primes are:

The number π_k(n) of positive integers less than or equal to n with exactly k prime divisors (not necessarily distinct) is asymptotic to:[4]

${\displaystyle \pi _{k}(n)\sim \left({\frac {n}{\log n}}\right){\frac {(\log \log n)^{k-1}}{(k-1)!}},}$

a result of Landau.[5] See also the Hardy–Ramanujan theorem.